0

I have a variable $z$ given by:

$z=\sum_{i=1}^n z_i$

where $z_i$ are random variables with $z_i \sim N\left(0,1\right)$. Then it will be $z \sim N(0,n)$ and the correlation between a variable $z_i$ and $z$ is $\frac{1}{\sqrt{n}}$.

Given now $y = e^{\alpha+\beta z}$, what is the correlation between one of the variables $z_i$ and $y$?

Addendum

Is my result correct?

$\frac{\beta e^{\frac{\beta^2}{2}}}{\sqrt{\left(e^{n\beta^2}-1 \right )e^{n\beta^2}}}$

mdewey
  • 16,541
  • 22
  • 30
  • 57
AttDefCon
  • 11
  • 3
  • 1
    This result is incorrect: as $|\beta|$ grows, the result's value rapidly approaches zero, whereas a scatterplot strongly suggests the correlation is noticeably nonzero. The method given in the near-duplicate indicated by @JarleTufto will give a correct solution. – whuber Nov 04 '19 at 15:44
  • 1
    I can see that @whuber. I think i also missed a factor of $n$ in the exponential of the numerator. – AttDefCon Nov 04 '19 at 16:14

0 Answers0